1

Comparing Edge Detection Methods based onStochastic Entropies and Distances for PolSAR

ImageryAbraao D. C. Nascimento, Michelle M. Horta, Alejandro C. Frery, Member, and

Renato J. Cintra, Senior Member

Abstract—Polarimetric synthetic aperture radar (PolSAR) hasachieved a prominent position as a remote imaging method.However, PolSAR images are contaminated by speckle noisedue to the coherent illumination employed during the dataacquisition. This noise provides a granular aspect to the image,making its processing and analysis (such as in edge detection)hard tasks. This paper discusses seven methods for edge detectionin multilook PolSAR images. In all methods, the basic ideaconsists in detecting transition points in the finest possible stripof data which spans two regions. The edge is contoured usingthe transitions points and a B-spline curve. Four stochasticdistances, two differences of entropies, and the maximum like-lihood criterion were used under the scaled complex Wishartdistribution; the first six stem from the h-φ class of measures. Theperformance of the discussed detection methods was quantifiedand analyzed by the computational time and probability ofcorrect edge detection, with respect to the number of looks,the backscatter matrix as a whole, the SPAN, the covariancean the spatial resolution. The detection procedures were appliedto three real PolSAR images. Results provide evidence that themethods based on the Bhattacharyya distance and the differenceof Shannon entropies outperform the other techniques.

Index Terms—Image analysis, information theory, polarimetricSAR, edge detection.

I. INTRODUCTION

POLARIMETRIC synthetic aperture radar (PolSAR) hasachieved a prominent position as a remote imaging

method [1]. SAR images are contaminated by speckle, a typ-ical noise present in data acquired with coherent illuminationsubject to multipath interference. Speckle noise introduces agranular aspect to the image, and its multiplicative naturemakes SAR image analysis a challenging task [2]. Most ofthe classical image processing methods assume additive noise,which is ineffective for processing SAR imagery [3]. Thus,PolSAR image analysis require specifically tailored signalprocessing techniques.

This work was supported by CNPq, Fapeal, FAPESP, and FACEPE, Brazil.A. D. C. Nascimento is with the Departamento de Estatıstica, Universidade

Federal de Pernambuco, Cidade Universitaria, 50740-540, Recife, PE, Brazil,e-mail: abraao@de.ufpe.br

M. M. Horta is with the Departamento de Computacao, UniversidadeFederal de Sao Carlos, Rod. Washington Luıs km 235, 13565-905, Sao Carlos,SP, Brazil, e-mail: michellemh@gmail.com

A. C. Frery is with the Instituto de Computacao, Universidade Federalde Alagoas, BR 104 Norte km 97, 57072-970, Maceio, AL, Brazil, e-mail:acfrery@gmail.com

R. J. Cintra is with the Signal Processing Group, Departamento de Es-tatıstica, Universidade Federal de Pernambuco, Cidade Universitaria, 50740-540, Recife, PE, Brazil, e-mail: rjdsc@ieee.org

Among image processing techniques, edge detection oc-cupies a central position. In simple terms, its purpose is toidentify boundaries between regions of different structuralcharacteristics [4]. In PolSAR systems, such features are usu-ally represented by different scattering characteristics, whichare effected by surface reflectance and speckle noise.

Several edge detection approaches have been proposed forSAR imagery, among them gradient-based methods [5]–[8].Such approach consists in using a sliding window to define ameasure map that highlights the edges, resembling the edgestrength map technique [5]. Then a thresholding step is appliedto perform the sought edge detection. In a different approach,Oliver et al. [6] proposed a maximum likelihood methodaiming at two goals: (i) detecting the presence of an edgewithin a window and (ii) determining accurately the position ofthe edge. Another line of research is the specific study of edgedetection based on physical properties from urban areas. As anexample, Baselice and Ferraioli [9] utilized Markov randomfields to model jointly the amplitude and interferometric phaseof two complex SAR images.

Statistical procedures, including active contour models, havealso been applied to edge detection on SAR images [3], [4],[10]–[13]. Giron et al. [13] compare seven edge detectorsfollowing the general idea proposed by Gambini et al. [3]:finding transition points in strips of data. This latter methodwas successfully employed in [12] and [13] for a comparativestudy using other strategies based on the same active contourapproach. In [4], Frery et al. extended it to multilook PolSARdata using the polarimetric GH distribution as model.

In the present work, we aim at extending the proposal ofGambini et al. [3]. In its original idea, this method consistsin forming strips of data around regions that span from thecentroid of the candidate region to points located outside theregion under consideration. Resulting strips are then submittedto a screening phase, where each one is scanned looking forthe point which defines two distinct regions under it basedon maximization criterion, such as likelihood function. Thepoint that satisfies a decision rule is called transition point.A pre-processing step may be necessary in order to definethe initial regions. A post-processing step defines the edges ofthe regions by combining the transition points with B-splinecurves.

For any type of approach, the edge detection problems canbe related to three main aspects: (i) the procedure for detection,(ii) the determination of the most accurate edge position,

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and (iii) the specification of the window size (the windowrepresents a square window or a strip of data). The lattermay influence all others aspects in such manner that smallerwindows may not convey enough information to identify thepresence of edges, while bigger windows may contain morethan two edges. Then, the ideal window size is the one thatprovides only a single edge within the window [6]. Followingthe approach discussed in [3], we assume that there is oneedge within the window provided by the initial selection.

Our proposal improves or extends those procedures in threesenses: (i) the finest possible strips of data are used, namelythe ones of one pixel width, and (ii) stochastic distances anddifference of entropies are employed as objective functionsto be maximized; not only the likelihood function as in [3].(iii) the influence of the spatial image resolution is performed.

The Kullback-Leibler and Bhattacharyya distances havebeen used by Morio et al. [14] as a scalar contrast measurebetween different channels of polarimetric and interferometricsynthetic aperture radar (PolInSAR) images. These distanceswere calculated assuming that PolInSAR images follow thecomplex multidimensional Gaussian circular distribution. Thedistances were then compared by their discrimination ability.In the present paper, this work extended in two ways: (i) thecomplex Wishart model is utilized instead of the Gaussiancircular distribution, and (ii) four distances, particular casesof the h-φ class of divergences, were considered as well astwo difference of entropies. Other information theory measureshave been employed for change detection. Erten et al. [15]proposed a method based on mutual information for PolInSARimages assuming the scaled complex Wishart law. As a result,they could verify that the new detector is more efficient thanthe one based on the maximum likelihood ratio statistic [16].

Therefore, the contribution of this paper is two fold. Firstly,the paper presents a detailed discussion about information the-ory measures (using h-φ divergences and entropies) as criteriafor the edge detection problem [3]. Secondly, we compare theirperformance by edge accuracy and computational time, usingcontrast and spatial resolution as factors. Three real multilookPolSAR images are employed to show the application of theproposal. The best results were obtained using Renyi andBhattacharya distances and difference of entropies.

This paper is structured as follows. Section II presentsthe seven assessed measures. An analysis of the proposedtechniques using real and simulated data is presented inSection III. To that end, Monte Carlo experiments are per-formed in Section III-A to assess required computational timeand probability of correct detection. Section III-B presentsapplications to real PolSAR images. Finally, the results aresummarized in Section IV.

II. EDGE DETECTION APPROACHES UNDER SCALEDCOMPLEX WISHART DISTRIBUTION

In this section, we present seven approaches for detectingedges in PolSAR images based on the methodology employedby [3], [4], [12], [13]: finding a transition point in a stripof data which is an estimator of the edge position. In allcases, the estimator is the maximum of a given function. The

first function is a likelihood, which is a multivariate extensionof the approach shown in [12]. Remaining functions are h-φdivergences and (difference of) entropies [17]–[19].

The edge detection methodologies considered in this paperoperate in multiple stages: (i) identifying the initial centroid ofthe area of interest in a automatic, semiautomatic, or manualmanner; (ii) casting rays from the centroid to the outside ofthe area; (iii) collecting data around the rays; (iv) detectingpoints in the strips of data which provide evidence of a changeof properties, a transition; (v) defining the contour using aimputation method among the transition points, such as B-Splines [3]. We bring contributions to stages (iii) and (iv).

Initially, let us admit a region R with centroid C. Rays aretraced from C to control points Pi, i = 1, 2, . . . , S, locatedoutsideR; the S resulting rays can be represented by segmentsin the form s(i) = CPi and the angle between consecutive raysis εi = ∠(s(i), s(i+1)). Rays are converted into pixels usingBresenham’s midpoint line algorithm [20]. This representationprovides the thinnest possible digital representation for a ray.This contrast with the 20-pixel wide strips employed in [3],[4], [12], [13]. This setup is illustrated in Fig. 1.

Fig. 1. Edge detection on the polarimetric strip s(i−1) from the centroidC of a region to the control point Pi−1.

Data are assumed to follow a scaled complex Wishartdistribution [21]. The scaled complex Wishart distribution hasdensity given by

fZ(Z ′;Σ, L) =LmL|Z ′|L−m

|Σ|LΓm(L)exp[−L tr

(Σ−1Z ′

)], (1)

where Z ′ is possible outcomes of Z, Σ represents the covari-ance matrix, L is the number of looks, m is the number of

3

polarization channels, Γm(L) = πm(m−1)/2∏L−1i=0 Γ(L − i)

is the multivariate gamma function, and | · | and tr(·) arethe determinant and the trace, respectively. We refer to thisdistribution as W(Σ, L)

The strip of data collected around the ith ray s(i), i =1, 2, . . . , S, contains N (i) pixels. Each pixel k of a given stripi is assumed to be described by the return matrix Z(i)

k , whichis distributed according to the scaled complex Wishart law [4].Denoting the correct boundary position as j(i), we have thefollowing configuration:{

Z(i)k ∼ W(Σ

(i)A , L

(i)A ), for k = 1, . . . , j(i),

Z(i)k ∼ W(Σ

(i)B , L

(i)B ), for k = j(i) + 1, . . . , N (i),

.

(2)In other words, each strip is formed by two patches of samples,and each patch obeys a complex Wishart law with differentparameter values. In this work, we assume that the numberof looks is estimated beforehand and remains constant forthe whole image. Thus, L(i)

A = L(i)B = L in every strip

i = 1, 2, . . . , S. This supposition is realistic, since the numberof looks is obtained from radar upon reception of backscatteredpulses [21].

The main idea is to estimate the edge position j(i) on thedata strip around ray s(i) according to a specified decision rule.Gray lines in Fig. 1 illustrate different configurations when theedge position is shifted.

The model stated in (2) assumes that at most one transitionoccurs in any given ray. Issues may arise if multiple transitionstake place; this is discussed further in Section III-B, Fig. 14.

In the following we present three different decision rules.For brevity, we omit the superscript index (i), since we focusour analysis on a single strip.

A. Log-likelihood

The likelihood function of the sample described by (2) isgiven by

L(j) =

j∏k=1

fZ(Z ′k;ΣA, L)

N∏k=j+1

fZ(Z ′k;ΣB , L),

where Z ′k is possible outcomes of the random matricesdescribed in (2). The log-likelihood is more convenient formaximization purposes:

`(j) = logL(j) =

j∑k=1

log fZ(Z ′k;ΣA, L)+

N∑k=j+1

log fZ(Z ′k;ΣB , L).

(3)The maximum likelihood estimator ML of the index at thelocation where the transition occurs is given by

ML = arg maxj`(j). (4)

Using the model given by (1) in (4), with minor algebraicmanipulation one obtains

`(j) =N[−mL(1− logL)− log Γm(L)

]+ L

[j log |ΣA(j)|

+ (N − j) log |ΣB(j)|], (5)

where ΣA(j) and ΣB(j) are the maximum likelihood esti-mators for ΣA and ΣB , respectively, with respect to edgeposition j. This estimator satisfies the following expression:

ΣI(j) =

{j−1

∑jk=1Zk if I = A,

(N − j)−1∑Nk=j+1Zk if I = B

B. Information Theory

Information theory measures have received considerableattention as tools for contrast quantification [22], [23]. Inthis section, we describe novel methodologies for detectingPolSAR boundaries based on such measures. We adopt thefollowing notation.

Let X and Y be two random matrices obeying the scaledcomplex Wishart distribution with same known numbers oflooks. The associated probability densities are fX(Z ′;θ1)and fY (Z ′;θ2), respectively, where θ1 = vec(ΣA) andθ2 = vec(ΣB) are their parameter vectors and vec(·) is thevectorization operator. Both densities are defined over the setof Hermitian positive definite matrices A.

1) Stochastic distances:(i) Kullback-Leibler:

dKL(X,Y ) =1

2

∫A

[fX(Z ′;θ1)− fY (Z ′;θ2)]

× logfX(Z ′;θ1)

fY (Z ′;θ2)dZ ′.

(ii) Renyi of order 0 < β < 1:

dRD-β(X,Y ) =1

β − 1

× log[∫A fX(Z ′;θ1)

βfY (Z ′;θ2)

1−βdZ ′

2

+

∫A fX(Z ′;θ1)

1−βfY (Z ′;θ2)

βdZ ′

2

].

(iii) Bhattacharyya:

dBA(X,Y ) = − log

∫A

√fX(Z ′;θ1)fY (Z ′;θ2) dZ ′.

(iv) Hellinger:

dH(X,Y ) = 1−∫A

√fX(Z ′;θ1)fY (Z ′;θ2) dZ ′.

In the above expressions, the differential element is given by

dZ ′ =

m∏i=1

dZ ′ii

m∏i, j = 1︸ ︷︷ ︸

i<j

d<{Z ′ij}d={Z ′ij},

where Z ′ij is the (i, j)th entry of matrix Z ′, and <{·} and={·} return the real and imaginary parts of their arguments,respectively. Provided that the densities characterize the samedistribution and possibly differ only in the parameter, we use(θ1,θ2) instead of (X,Y ) as arguments of the considereddistances. Frery et al. [24] derived analytical expressions forthe above distances considering Wishart distributions for thegeneral case ΣA 6= ΣB and LA 6= LB . Since we assume that

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the equivalent number of looks is the same, the distances canbe expressed according to:(i) Kullback-Leibler:

dKL(θ1,θ2) = L

[tr(Σ−11 Σ2 + Σ−12 Σ1)

2−m

].

(ii) Renyi:

dRD−β(θ1,θ2) =log 2

1− β+

1

β − 1log{

[ |(βΣ−11 + (1− β)Σ−12 )−1||Σ1|β |Σ2|1−β

]L+[ |(βΣ−12 + (1− β)Σ−11 )−1|

|Σ1|(1−β)|Σ2|β]L}

.

(iii) Bhattacharyya

dBA(θ1,θ2) =L

[log |Σ1|+ log |Σ2|

2

− log

∣∣∣∣(Σ−11 + Σ−12

2

)−1∣∣∣∣].(iv) Hellinger

dH(θ1,θ2) = 1−

[∣∣2−1(Σ−11 + Σ−12 )−1∣∣√

|Σ1||Σ2|

]L.

For L = 1, the Kullback-Leibler and Bhattacharyya distancesbecome similar to the expressions derived by Morio et al. [14].

Above expressions are modified according to [18] in orderto obtain test statistics with known asymptotic properties. Thustest statistics based on these distances can be readily derivedfor the null hypothesis H0 : θ1 = θ2:

SD(θ1(j), θ2(N − j)

)=

2j(N − j)vDN

dD(θ1(j), θ2(N − j)

),

(6)where D ∈ {KL, RD-β, BA, H}, vD ∈ {1, β−1, 4, 4}, respec-tively, θ1(j) = vec(ΣA(j)) and θ2(N − j) = vec(ΣB(N −j)) are estimators for θ1 and θ2 using random samples ofsizes j and N − j as well as , respectively.

Let p be the number of elements of the parameter vectors,i.e. p = m2, as discussed in [25]. As derived by Frery etal. [24], the test statistics given in (6) are asymptotically χ2-distributed with p degrees of freedom. This result holds trueas long as the maximum likelihood estimators are used withlarge samples [18].

Thus, four novel detectors can be proposed for finding edgeson PolSAR imagery by seeking for the point that maximizesthe test statistics between the two models, i.e.,

D = arg maxjSD(θ1(j), θ2(N − j)

)︸ ︷︷ ︸≡ SD(j)

= arg maxjSD(j),

where D = {KL,RD-β,BA,H}.2) Stochastic entropies: The concepts of “information” and

“entropy” received its fundamental mathematical treatment inthe context of data communications by Shannon in 1948 [26].Thenceforth, the proposition and the application of thesemeasures have become active research fields in several areas.

In particular, Frery et al. [27] derived entropies and associatedtest statistics for the scaled complex Wishart model.

As suggested by Pardo et al. [28], we consider the Renyiof order 0 < β < 1 and Shannon entropies, denoted byHR-β and HS, respectively. A discrimination statistics basedon stochastic entropies can be defined as

SM(θ1(j), θ2(N − j)) =j(HM(θ1(j))− v

)2σ2M(θ1(j))

+(N − j)

(HM(θ2(N − j))− v

)2σ2M(θ2(N − j))

,

(7)

where HM ∈ {HS, HR-β}, θi = [θi1 θi2 · · · θip]>,

i = 1, 2, θik is the kth element of vector θi, k = 1, 2, . . . , p,σ2M(θi) = δ∗iK(θi)

−1δi, the superscript > is the transpositionoperator, ∗ represents the conjugate transposition, K(θi) =E{−∂2 log fZ(Z;θi)/∂θ

2i } is the Fisher information matrix,

δi = [δi1 δi2 · · · δip]> such that δik = ∂HM(θi)/∂θik, p isthe size of vector θi, and

v =

[j

σ2M(θ1(j))

+(N − j)

σ2M(θ2(N − j))

]−1

×

[jHM(θ1(j))

σ2M(θ1(j))

+(N − j)HM(θ2(N − j))

σ2M(θ2(N − j))

].

Frery et al. [27] showed that the expression given in (7)follows asymptotically a χ2

1 distribution, and it can be used totest whether two samples from the scaled Wishart distributionpossess the same entropy, i.e., H0 : HM(θ1) = HM(θ2).Additionally, the Shannon and Renyi of order β entropies aregiven by, respectively:

HS(X) ≡ HS(θi) =−∫AfX(Z ′;θi) log fX(Z ′;θi)dZ

′

= E{− log fX(Z;θi)}

and

HR-β(X) ≡ HR-β(θi) =(1− β)−1 log

∫AfβX(Z ′;θi)dZ

′

=(1− β)−1 log E{fβ−1X (Z;θi)

},

for i = 1, 2. Applying these expressions to the density givenin (1) the following entropies and variances are obtained:• Shannon:

HS(Σ, L) =m(m− 1)

2log π −m2 logL+m log |Σ|

+mL+ (m− L)ψ(0)m (L) +

m−1∑k=0

log Γ(L− k)

and

σ2S =

[(m− L)ψ

(1)m (L) +m− m2

L

]2ψ(1)m (L)− m

L

+m2

Lvec(Σ−1

)∗(Σ⊗Σ

)vec(Σ−1

),

5

where ⊗ is the Kronecker product, “vec” is the operatorwhich vectorizes its argument, and ψ(v)

m is the vth-ordermultivariate polygamma function given by [25]

ψ(v)m (x) =

m−1∑i=0

ψ(v)(x− i),

and the ordinary polygamma function is [25]

ψ(v)(x) =∂v+1 log Γ(x)

∂xv+1,

for v ≥ 0.• Renyi of order 0 < β < 1: Denoting q = L+(1−β)(m−L), the Renyi entropy is expressed by

HR-β(ΣI , L) =m(m− 1)

2log π −m2 logL+m log |ΣI |

+

∑m−1i=0

[log Γ(q − i)− β log Γ(L− i)

]1− β

− mq log β

1− β

and

σ2R-β =

{β

1−β[ψ(0)m (q)− ψ(0)

m (L)]− mβ log(β)

1−β − m2

L

}2

ψ(1)m (L)− m

L

+m2

Lvec(Σ−1

)∗(ΣI ⊗Σ

)vec(Σ−1

).

These and other results are further discussed in [27].Therefore, two additional detectors can be defined by the

points that maximize the test statistics based on entropiesbetween the two distributions:

M = arg maxjSM(θ1(j), θ2(N − j))︸ ︷︷ ︸

≡ SM(j)

= arg maxjSM(j),

where M = {S,RE-β}.The expressions given in (6) and (7) have known asymptotic

distributions so, besides being discriminatory measures, theyalso have significance levels. This can be useful when thestrip of data conveys a single class and, therefore, no detectionshould be made. Nevertheless, in the subsequent section theseproperties are not considered, due to the fact that the procedureis bounded to use small samples.

III. APPLICATIONS

We now apply the methodologies presented in the previoussection to simulated and real data. Initially, in Section III-A,complex Wishart distributed scenarios were generated in orderto quantify and to assess the accuracy and computational loadof the detailed edge detection schemes. For such estimatesof the detection error as well as the execution time for eachmethod were compared. Finally, three applications to real dataare performed in Section III-B. These applications illustratethe difficulties of determining an ideal window size.

A. Precision and execution time

In order to measure and compare the accuracy of thediscussed seven edge detection techniques, we follow the samemethodology proposed by Frery et al. [4]. We estimate theprobability of detecting the edge with an error less than kpixels, for k = 1, 2, . . . , 10. This probability was estimatedusing strips of data of 400 pixels divided in halves, filledwith samples from the Wishart distribution with two differentcovariance matrices:

ΣA =

962892 19171− 3579i −154638 + 191388i56707 −5798 + 16812i

472251

(8)

ΣB =

360932 11050 + 3759i 63896 + 1581i98960 6593 + 6868i

208843

, (9)

which were observed by Frery et al. [4] in urban and forestimagery, respectively. In all cases four looks (L = 4) wereconsidered. Fig. 2 shows a simulated PolSAR image with thisconfiguration according to the Pauli decomposition [29].

Fig. 2. False color of the Pauli decomposition of a simulated image fromthe scaled Wishart distribution with four looks and two halves defined by ΣA

and ΣB .

Figure 3 shows the functions to be maximized as func-tions of the position for a typical simulation: the likelihood(Fig. 3(a)), distances (Fig. 3(b)), and entropies (Fig. 3(c)).The edge is at 200, and this position identified with a verticalsolid line; the symmetric interval of twenty pixels around it isshown in vertical dash-dot lines. All of these distances exhibita maximum which is close to the true edge position.

We generated 1000 independent scene simulations to obtainthe estimated boundaries positions (b(r) ∈ {ML, D, M},1 ≤ r ≤ 1000). The distance between these points and thetrue boundary (the absolute empirical bias) was evaluated foreach replication:

E(r) = |b(r)− 200|,

and the probability of observing an error smaller than a certainnumber of pixels is estimated by relative frequencies as

f(k) = T (k)/1000,

where T (k) the number of replications for which the error issmaller than k pixels.

Fig. 4 presents, in semilogarithmic scale, the estimatedprobability of finding the edge with an error equal or smallerthan k pixels, k = 1, 2, . . . , 10. Results show evidence that themethods based on entropies have the best results for k ≤ 3.On the other hand, admitting an error k ≥ 4, the methodinvolving stochastic distances have the higher estimated detec-tion probability. This fact suggests that a combination of these

6

(a) Likelihood detector (b) Distance detectors

(c) Entropy detectors

Fig. 3. Illustration of the proposed edge detection measures on imageswith halves which followW(ΣA, 4) andW(ΣB , 4) distributions. The solidvertical line is at the edge and the dashed lines are at 10 pixels from it.

methodologies could lead to even more precise edge detection.Finally, the method based on the likelihood is outperformedby those based on stochastic distances; however, the formermethod provides better estimates when compared to SS andSR-β .

Fig. 4. Performance of detectors on phantom images with two halvesdistributed according to scaled complex Wishart laws.

1) Influence of the backscatter matrix and time execution:Now we devise a simulation study where data is more similarlydistributed; therefore offering a more difficult scenario. Alsowe aim at assessing the execution time of each method. Stripsof data were defined with halves which follow distributionsW((1 + k)ΣB , 4) and W(ΣB , (1 + v)4), respectively. Thissituation is referenced by “case-(k, v)”. Fig. 5 presents imagessampled in cases (0, 0), (1/2, 0), (1, 0), (3, 0), and (4, 0),i.e., with the same number of looks and varying covariancematrices.

Fig. 5. Images generated from W (ΣB ·(1+k); 4), for k = 0, 1/2, 1, 3, 4,mapping [HH+VV] – [HV] – [HH−VV] onto the RGB channels.

The precision and execution time of the procedure wereestimated using one thousand replications. The latter wasmeasured in seconds of CPU cycles, while for the formerwe adopted a “hit-and-miss” criterion: a run is consideredsuccessful if it finds the true boundary, otherwise it is assumedwrong. The mean number of successful runs are reported inFig. 6.

Fig. 6 and Table I present the average estimates of precisionfor all methods under several situations; the first column ofTable I shows the label for each case which is used in Figs. 6and 7. The best estimators are highlighted in boldface.

Situations 1 to 4, which correspond to cases where thecovariance matrix does not vary or changes by a small amount,the hardest ones to perform edge detection. Thus they oftenlead to low precision regardless the technique. It is noticeablethat entropy methods consistently outperform other techniquesin these particularly challenging situations. The performanceof all procedures is comparable at remaining situations. Wecould identify that the maximum likelihood and entropy-basedestimators excelled, in three and in two situations, respectively,but for a small difference of the value of the true boundary.

TABLE IESTIMATES FOR BOUNDARY POINTS (ESTIMATES CLOSEST TO THE TRUE

EDGE ARE HIGHLIGHTED)

Label Case ML KL BA H RD-0.8 S = RE-0.8

1 (0, 3) 17.662 16.870 17.288 18.871 17.007 22.5822 (0, 7) 18.531 17.609 18.074 19.994 17.816 26.1983 (0, 1) 23.834 23.306 23.843 25.604 23.538 36.2874 (0.1, 1) 27.746 28.688 29.176 30.806 28.991 44.6265 (0.1, 0) 49.726 54.055 53.877 52.653 53.961 55.3586 (1, 3) 50.738 50.826 50.841 50.842 50.834 50.8447 (2, 7) 50.941 50.966 50.971 50.974 50.968 50.9708 (1, 0) 50.952 51.129 51.119 51.102 51.120 51.1109 (2, 0) 50.979 51.052 51.048 51.042 51.050 51.052

7

Fig. 6. Precison of the detection on polarimetric images with halves whichfollow W((1 + k)ΣB , 4) and W(ΣB , (1 + v)4) distributions.

Fig. 7 presents the mean execution time of each method inseconds. Fig. 7(a) presents obtained detection times, where itis noticeable that the maximum likelihood method associatedto ML is the slowest procedure. Fig. 7(b) shows only thetimes required by procedures based on stochastic distances.We notice that the method based on the Kullback-Leiblerdivergence, which furnishes KL, was the fastest one. Theresults can be summarized in the following inequalities of therequired computational times t for each technique:

tML ≥ tRD ≥ tRE ≥ tS ≥ tH ≥ tBA ≥ tKL .

The influence of the case on the computing time is very small.All implementations are in the R version 2.13.2 programminglanguage, running on a PC with an Intel© Core® i7-2630QM2.00GHz with 4 GB RAM.

(a) All times (b) Information theoretic tool times

Fig. 7. Detection times of the proposed methods in terms of the situationspresented in Table I.

2) Influence of the backscatter matrix SPAN and imageresolution: In the following we assess the relative information

provided by the polarimetric data with respect to single-channel data for the problem of finding edges. Assuming thescaled complex Wishart with parameters Σ and L for thepolarimetric data, each intensity channel is described by thegamma distribution with density given by

fZi(Z ′i;L/σ

2i , L) =

LLZ ′iL−1

σ2Li Γ(L)

exp(−LZ ′i/σ2

i

),

for i ∈ {HH,HV,VV}, where σ2i is the (i, i)th entry of Σ

and Z ′i is the (i, i)th entry of the random matrix Z [30]. Thisdensity can be used in the log-likelihood of strips (cf. (3)) toderive the estimator of the edge position (cf. (4)).

We considered two random matrices X ∼ W(ΣB , 4) andY ∼ W(ΣB′ , 4):• Case A: diag(ΣB′) = (1 + δ) diag(ΣB) and

adiag(ΣB′) = adiag(ΣB), where diag(A) =[A11A22 . . . Amm]>, for a square matrix A with orderm, and adiag(A) = [w>w∗]>, w = [A12 . . . A1m

A23 . . . A2m . . . A(m−1)m]>. This case represents theinfluence of the SPAN [31], i.e., the trace of backscattermatrix when δ ≥ 0.

• Case B: diag(ΣB′) = diag(ΣB) and adiag(ΣB′) =adiag(ΣB) + (1 + δ){<[adiag(ΣB)] + =[adiag(ΣB)]}.This aims at quantifying the influence of the covariancebetween polarization channels varying the values of δ ≥−1.

A Monte Carlo simulation was performed to quantify theperformance of the edge detectors under these conditions.

For Case A, 1000 images with 100 pixels in each halfwere generated with δ ∈ {0.2, 0.4, 1.0}. Edge detection wasperformed, and the probability of detecting within an errorequal or smaller than k ∈ {1, 2, . . . , 10} pixels was estimated.Fig. 8 presents the results. As expected, all polarimetricmethods outperformed the methods based on single-channeldata. Additionally, these results provide evidence that theperformance of the methods tends to be similar when theSPAN is significantly different, i.e., when the halves arequite different. Moreover, boundary detection based on entropymeasures outperformed all other methods in Case A.

Fig. 9 shows the performance of the discussed methods inCase B, for δ ∈ {0.2, 0.3, 1.0}. As in Case A, polarimetricdata led to better results than single-channel images. In thiscase, detectors based on stochastic distances and maximumlikelihood outperformed those based on entropies. Fig. 9(a)legend presents the detectors ordered from best to worst fromtop to bottom and from left to right. The best result wasobtained by ML, but small differences were observed amongmethods involving distances.

We now analyze the effect of spatial resolution on theprecision of the edge detection procedures. To that end, wesimulated a strip of 200 observations divided in two halves,each with different distributions, namely X ∼ W(ΣB , 4) andY ∼ W(ΣB′ , 4), such that diag(ΣB′) = 1.2 diag(ΣB), withΣB defined in (9). This strip is assumed to have the best spatialresolution, and is denoted “1 ÷ 1”. The position of the edge,which is at the middle of the strip, i.e. at 100, is estimatedwith all the available techniques. Then the spatial resolution of

8

(a) δ = 0.2 (b) δ = 0.4

(c) δ = 1.0

Fig. 8. Influence of the SPAN on the probability of detecting and edge withan error equal or smaller than a certain number of pixels.

(a) δ = 0.2 (b) δ = 0.4

(c) δ = 1.0

Fig. 9. Influence of the covariance between polarization channels on theprobability of detecting and edge with an error equal or smaller than a certainnumber of pixels.

the strip is degraded by taking the mean of pairs of contiguousobservations and downsampling one each two pixels obtaining,thus, a 1 ÷ 2 resolution image. The estimation of the edgeposition, which is now at 50, is performed. A new pyramid isformed, denoted “1÷ 4”, and the estimation of the edge (nowat 25) is again performed. The equivalent number of looks

L is assumed know, and at each resolution takes the values4, 8 and 16. This procedure was repeated generating 1000independent initial strips, and the bias (B, the expected valueof the estimator minus the true value), the standard deviation(sd), the coefficient of variation (CV) and the mean squarederror (MSE) of each estimator •) were estimated. The resultsare shown in Table II.

The detection bias was always very small, being the largestvalues in the order of 10−2 of a pixel and the smallest onesin the order of 10−3 of a pixel. The absolute bias tendedto exhibit a small increase when the spatial resolution wasreduced, while the standard deviation and the mean squarederror were reduced. The width of the Gaussian symmetric con-fidence intervals at the 95% confidence level ranges between,approximately, half a pixel and one pixel, rendering, thus,that all techniques exhibit comparable accuracy. Changing theresolution had little effect on the coefficient of variation.

B. Application to real dataWe applied the methods to three real images. The first

example describes a simple situation: well separated regionswith most of the strips including two regions only. The secondand third examples present more sophisticated problems dueto the size of the strips (larger strips may span more than tworegions).

The first application was performed on a San Francisco Bayimage obtained by the AIRSAR sensor in L-band with fournominal looks, which is available in [32]. Fig. 10(a) shows a150×150 scene of the image with three well defined regions:ocean (dark area – top and left part of the image), forest (grayarea – top and right), and urban area (light area – bottom).Figs. 10 and 11 present the result of detecting the edgeswhich separate urban and ocean regions from other classes,respectively. The following analysis was performed by visualinspection on edges reconstructed from the estimated transitionpoints and B-splines of fourth degree.

In Fig. 10(b), estimator KL presents two mildly biasedboundaries (highlighted with green squares), whereas S andRE-0.8 present only one. Remaining methods present similarbehavior.

Fig. 11(b) also exhibits three situations where the detectededge was slightly biased, namely KL, RE-0.8 and H, eachwith one noticeable deviation. Again, other discussed methodsbehave alike.

The Bhattacharyya and likelihood techniques did not missany detection in these examples. This, and the simulationresults on accuracy and computational time, leads us to suggestB as the fastest and most efficient detector.

Fig. 12 shows an E-SAR image over the surroundings ofMunich, Germany, which was obtained with 3.2 (equivalentnumber of) looks. The nonconvex region to the center ofFig. 12(a) is of interest, for which an arbitrarily centroidand thirteen rays were defined. Fig. 12(b) summarizes theresults. It shows the edges, and the label of the ray forwhich the transition point was identified far from the visualtrue point. Both maximum likelihood and Kullback-Leiblermethods failed at detecting only one situation, highlighted witha green square. The other results are similar.

9

TABLE IIEFFECT OF THE SPATIAL RESOLUTION ON THE EDGE DETECTION

Boundary detection methods

Single channel data Full PolSAR data

Resolution Measures HH-ML HV-ML VV-ML ML KL BA H RD-0.8 S = RE-0.8

1÷ 1 100B 0.502 −0.547 2.866 0.273 −0.793 −0.785 −0.545 −0.792 −0.171sd(•) 52.850 48.948 50.683 18.388 24.338 22.733 18.826 24.338 15.028CV(•) 0.263 0.246 0.246 0.092 0.123 0.115 0.095 0.123 0.075MSE(•) 2791.361 2394.661 2599.036 338.076 594.280 518.758 355.249 594.232 225.726

1÷ 2 1.541 −0.252 3.335 0.808 −0.151 −0.153 −0.052 −0.152 0.42025.435 22.977 25.061 8.984 9.880 9.875 9.406 9.875 7.3730.250 0.230 0.243 0.089 0.099 0.099 0.094 0.099 0.073

648.653 527.482 638.523 81.288 97.549 97.433 88.386 97.434 54.486

1÷ 4 2.144 1.190 4.504 1.876 0.784 0.932 0.990 0.890 1.36212.037 11.235 11.790 4.451 4.933 4.713 4.671 4.737 3.6030.236 0.222 0.226 0.087 0.098 0.093 0.092 0.094 0.071

145.882 126.445 143.948 20.670 24.468 22.404 22.039 22.617 13.435

(a) Urban area vs. sea and forest

(b) Detected egdes

Fig. 10. Performance of detectors with centroid on urban region fromAIRSAR image of San Francisco Bay. Errors highlighted in green boxes.

There are practical situations in which all methods performalike. Consider the EMISAR image of the agricultural areaof Foulum, Denmark, presented in Fig. 13. This image wasobtained in L-band and quad-pol; its HH channel is shown in

(a) Sea vs. urban area and forest

(b) Detected egdes

Fig. 11. Performance of detectors with centroid on ocean region fromAIRSAR image of San Francisco Bay. Errors highlighted in green boxes.

Fig. 13(a). Notice that the method based on the joint likelihood(Fig. 13(b)) works similarly to BA and RD-0.8 (Fig. 13(c)),RE-0.8 (Fig. 13(d)), and KL (Fig 13(e)). However, the besttechnique regarding execution times is the one based on the

10

(a) HH channel of the ESAR image with selected rays

(b) Detection in actual images

Fig. 12. Edge detection in an E-SAR image over the surroundings of Munich,Germany. Error highlighted in the green box.

Bhattacharyya distance.In order to analyze further the result obtained by the

Hellinger distance, presented in Fig. 13(f), consider the rayzoomed in Fig. 14(b), and notice that it crosses three regions:gray, white and black leading, therefore, to two possibleedges. As all the techniques are proposed in terms of themaximization of a discrimination measure, it is expectedthat the estimated boundary is the position of the highestdistinction between two regions. In this case, it takes placein the second edge, i.e., between the white and black areas.This is confirmed by Fig. 14(c), which shows the likelihood,but not in Fig. 14(d), which shows the Hellinger distance. Allother criteria behave alike, as shown in Fig. 14(e).

The Hellinger distance is limited to the [0, 1] interval. Addi-tionally, let v(k) = dH(θ1,θ2 + k), empirical results provideevidence that there is a subset Θ of the parametric space ofthe scaled complex Wishart distribution for which v(k) ≈ 1for all k ∈ Θ. This fact renders the statistics SH(θ1,θ2 + k)inappropriate for quantifying distinctions between distributedWishart random fields when k ∈ Θ.

This behavior opens new research lines based on a redefini-tion of the proposed methods in order to cope with situationsin which rays cross three or more distinct regions.

IV. CONCLUSIONS

In this paper, boundary detection procedures using infor-mation theory measures (stochastic distances and entropies)

(a) Real Images with selected axes (b) ML

(c) BA = RD-0.8 (d) RE-0.8 = S

(e) KL (f) H

Fig. 13. Edge detection of an agricultural parcel in an EMISAR image ofFoulum.

and a likelihood function were considered. Due to its analytictractability, these techniques are derived under the assumptionthat the Wishart distribution is an acceptable model for fullPolSAR data. In order to quantify and compare their perfor-mances, phantoms and real PolSAR images were used.

The performance of each procedure was quantified by theprobability of correctly detecting the edge position within atmost k pixels, as well as by the execution time. Problemswith different levels of complexity were assessed in order toquantify the influence of the number of looks, the backscat-ter matrix as a whole, the SPAN, the covariance, and thespatial resolution. Also, the information provided by the fullpolarimetric format with respect to each intensity channel wasassessed, leading to the conclusion that the complete data set isrecommended for edge detection. The results provide evidencethat all methods are able to detect edges correctly withinfour pixels, with the exception of entropy-based techniqueswhich fail in more complex scenarios with similar covariance

11

(a) A ray crossing three different regions (b) Zoom

(c) ML (d) H

(e) •

Fig. 14. Details of of the edge detection in a hard-to-deal-with configurationfrom the EMISAR image of Foulum.

matrices.The discriminatory function based on the Kullback-Leibler

distance is the fastest to compute, while the likelihood functionis the slowest one among the considered measures.

The simulation studies revealed that information theorymeasures are consistently closer to the true boundary than thejoint likelihood detector. In particular, Shannon and Kullback-Leibler detectors provided the most accurate detection and thesmallest execution time, respectively.

Finally, three applications to real data obtained by E-SAR, EMISAR, and AIRSAR sensors were performed. Theseemingly poor performance of the edge detection based onthe Hellinger distance was identified and explained by itsproperties.

As a conclusion, techniques which employ Renyi and Bhat-tacharyya distances and entropies outperform other methods

and are recommended for edge detection in PolSAR imagery.Further research aims at considering models which include

heterogeneity [4], [21], [33], robust, improved and nonpara-metric inference [34]–[37], and small samples techniques [38].

ACKNOWLEDGMENT

The authors are grateful to CNPq, Facepe, Fapeal, FAPESP,and Capes for their support.

REFERENCES

[1] J. S. Lee and E. Pottier, Polarimetric Radar Imaging: From Basics toApplications. Boca Raton: CRC, 2009.

[2] C. Oliver and S. Quegan, Understanding Synthetic Aperture RadarImages, ser. The SciTech radar and defense series. SciTech Publishing,1998.

[3] J. Gambini, M. Mejail, J. Jacobo-Berlles, and A. C. Frery, “Featureextraction in speckled imagery using dynamic B-spline deformablecontours under the G0 model,” International Journal of Remote Sensing,vol. 27, no. 22, pp. 5037–5059, 2006.

[4] A. C. Frery, J. Jacobo-Berlles, J. Gambini, and M. Mejail, “PolarimetricSAR image segmentation with B-splines and a new statistical model,”Multidimensional Systems and Signal Processing, vol. 21, pp. 319–342,2010.

[5] R. Touzi, A. Lopes, and P. Bousquet, “A statistical and geometricaledge detector for SAR images,” IEEE Transactions on Geoscience andRemote Sensing, vol. 26, no. 6, pp. 764–773, 1988.

[6] C. J. Oliver, D. Blacknell, and R. G. White, “Optimum edge detection inSAR,” IEE Proceedings Radar, Sonar and Navigation, vol. 143, no. 1,pp. 31–40, 1996.

[7] R. Fjortoft, A. Lopes, P. Marthon, and E. Cubero-Castan, “An optimalmultiedge detector for SAR image segmentation,” IEEE Transactions onGeoscience and Remote Sensing, vol. 36, no. 3, pp. 793–802, 1998.

[8] F. Xingyu, Y. Hongjian, and F. Kun, “A statistical approach to detectedges in SAR images based on square successive difference of averages,”IEEE Geoscience and Remote Sensing Letters, vol. 9, no. 6, pp. 1094–1098, 2012.

[9] F. Baselice and G. Ferraioli, “Statistical edge detection in urban areasexploiting SAR complex data,” IEEE Geoscience and Remote SensingLetters, vol. 9, no. 2, pp. 185–189, 2012.

[10] M. Horrit, “A statistical active contour model for SAR image segmen-tation,” Image and Vision Computing, vol. 17, no. 3-4, pp. 213–224,1999.

[11] O. Germain and P. Refregier, “Edge location in SAR images: Perfor-mance of the likelihood ratio filter and accuracy improvement withan active contour approach,” IEEE Transactions on Image Processing,vol. 10, no. 1, pp. 72–78, 2001.

[12] J. Gambini, M. Mejail, J. Jacobo-Berlles, and A. C. Frery, “Accuracyof edge detection methods with local information in speckled imagery,”Statistics and Computing, vol. 18, no. 1, pp. 15–26, 2008.

[13] E. Giron, A. C. Frery, and F. Cribari-Neto, “Nonparametric edge detec-tion in speckled imagery,” Mathematics and Computers in Simulation,vol. 82, pp. 2182–2198, 2012.

[14] J. Morio, P. Refregier, F. Goudail, P. C. Dubois Fernandez, andX. Dupuis, “A characterization of Shannon entropy and Bhattacharyyameasure of contrast in polarimetric and interferometric sar image,”Proceedings of the IEEE (PIEEE), vol. 97, no. 6, pp. 1097–1108, June2009.

[15] E. Erten, A. Reigber, L. Ferro-Famil, and O. Hellwich, “A new coherentsimilarity measure for temporal multichannel scene characterization,”IEEE Transactions on Geoscience and Remote Sensing, vol. 50, no. 7,pp. 2839–2851, july 2012.

[16] K. Conradsen, A. A. Nielsen, J. Schou, and H. Skriver, “A test statistic inthe complex Wishart distribution and its application to change detectionin polarimetric SAR data,” IEEE Transactions on Geoscience andRemote Sensing, vol. 41, no. 1, pp. 4–19, 2003.

[17] A. C. Frery, A. D. C. Nascimento, and R. J. Cintra, “Information theoryand image understanding: An application to polarimetric SAR imagery,”Chilean Journal of Statistics, vol. 2, no. 2, pp. 81–100, 2011.

[18] M. Salicru, M. L. Menendez, L. Pardo, and D. Morales, “On the ap-plications of divergence type measures in testing statistical hypothesis,”Journal of Multivariate Analysis, vol. 51, pp. 372–391, 1994.

12

[19] M. Salicru, M. L. Mendendez, and L. Pardo, “Asymptotic distribution of(h, φ)-entropy,” Communications in Statistics - Theory Methods, vol. 22,no. 7, pp. 2015–2031, 1993.

[20] J. J. McConnell, Computer Graphics: Theory into Practice. Jones andBartlett, 2006.

[21] C. C. Freitas, A. C. Frery, and A. H. Correia, “The polarimetric Gdistribution for SAR data analysis,” Environmetrics, vol. 16, no. 1, pp.13–31, 2005.

[22] A. D. C. Nascimento, R. J. Cintra, and A. C. Frery, “Hypothesistesting in speckled data with stochastic distances,” IEEE Transactionson Geoscience and Remote Sensing, vol. 48, no. 1, pp. 373–385, 2010.

[23] N. Ebrahimi, E. S. Soofi, and R. Soyer, “Information measures inperspective,” International Statistical Review, 2010.

[24] A. C. Frery, A. D. C. Nascimento, and R. J. Cintra, “Analytic expressionsfor stochastic distances between relaxed complex Wishart distributions,”IEEE Transactions on Geoscience and Remote Sensing, in press.

[25] S. N. Anfinsen, A. P. Doulgeris, and T. Eltoft, “Estimation of the equiv-alent number of looks in polarimetric synthetic aperture radar imagery,”IEEE Transactions on Geoscience and Remote Sensing, vol. 47, no. 11,pp. 3795–3809, 2009.

[26] C. E. Shannon, “A mathematical theory of communication,” Bell SystemTechnical Journal, vol. 27, pp. 379–423, July 1948.

[27] A. C. Frery, R. J. Cintra, and A. D. C. Nascimento, “Entropy-basedstatistical analysis of PolSAR data,” IEEE Geoscience and RemoteSensing, in press.

[28] L. Pardo, D. Morales, M. Salicru, and M. L. Menendez, “Large samplebehavior of entropy measures when parameters are estimated,” Commu-nications in Statistics - Theory and Methods, vol. 26, no. 2, pp. 483–501,1997.

[29] S. Maitra, M. G. Gartley, and J. P. Kerekes, “Relation between degreeof polarization and Pauli color coded image to characterize scatteringmechanisms,” in Society of Photo-Optical Instrumentation Engineers(SPIE), 2012.

[30] M. Hagedorn, P. J. Smith, P. J. Bones, R. P. Millane, and D. Pairman,“A trivariate chi-squared distribution derived from the complex Wishartdistribution,” Journal of Multivariate Analysis, vol. 97, no. 3, pp. 655–674, 2006.

[31] F. Cao, W. Hong, Y. Wu, and E. Pottier, “An unsupervised segmentationwith an adaptive number of clusters using the SPAN/H/α/A space andthe complex Wishart clustering for fully polarimetric SAR data anal-ysis,” IEEE Transactions on Geoscience and Remote Sensing, vol. 45,pp. 3454–3467, 2007.

[32] E. Pottier, L. Ferro-Famil, S. Allain, S. Cloude, I. Hajnsek, K. Pap-athanassiou, A. Moreira, M. L. Williams, A. Minchella, M. Lavalle,and Y.-L. Desnos, “Overview of the PolSARpro v4.0 software. Theopen source toolbox for polarimetric and interferometric polarimetricSAR data processing,” in IEEE International Geoscience and RemoteSensing Symposium, IGARSS 2009. University of Cape Town, CapeTown, South Africa: IEEE, July 2009, pp. 936–939.

[33] A. C. Frery, A. H. Correia, and C. C. Freitas, “Classifying multifre-quency fully polarimetric imagery with multiple sources of statisticalevidence and contextual information,” IEEE Transactions on Geoscienceand Remote Sensing, vol. 45, pp. 3098–3109, 2007.

[34] H. Allende, A. C. Frery, J. Galbiati, and L. Pizarro, “M-estimators withasymmetric influence functions: the GA0 distribution case,” Journal ofStatistical Computation and Simulation, vol. 76, no. 11, pp. 941–956,2006.

[35] E. Giron, A. C. Frery, and F. Cribari-Neto, “Nonparametric edge detec-tion in speckled imagery,” Mathematics and Computers in Simulation,vol. 82, pp. 2182–2198, 2012.

[36] M. Silva, F. Cribari-Neto, and A. C. Frery, “Improved likelihoodinference for the roughness parameter of the GA0 distribution,” En-vironmetrics, vol. 19, no. 4, pp. 347–368, 2008.

[37] K. L. P. Vasconcellos, A. C. Frery, and L. B. Silva, “Improvingestimation in speckled imagery,” Computational Statistics, vol. 20, no. 3,pp. 503–519, 2005.

[38] A. C. Frery, F. Cribari-Neto, and M. O. Souza, “Analysis of minutefeatures in speckled imagery with maximum likelihood estimation,”EURASIP Journal on Applied Signal Processing, vol. 2004, no. 16, pp.2476–2491, 2004.

Abraao D. C. Nascimento holds B.Sc. M.Sc.and D.Sc. degrees in Statistics from UniversidadeFederal de Pernambuco (UFPE), Brazil, in 2005,2007, and 2012, respectively. In 2013, he joined theDepartment of Statistics at Universidade Federal daParaıba. His research interests are statistical infor-mation theory, inference on random matrices, andasymptotic theory.

Michelle M. Horta received the B.S. degree in com-puter sciences from Universidade Catolica de Per-nambuco, Recife, Brazil, in 2002, the M.Sc. degreein computer sciences from Universidade Federal dePernambuco, Recife, Brazil, in 2004, and the Ph.D.degree in applied physics from Universidade de SaoPaulo, Sao Carlos, Brazil, in 2009. She is currentlyworking toward the Postdoctoral Research in theDepartamento de Computacao, Universidade Federalde Sao Carlos, Sao Carlos, Brazil. Her researchinterests are remote sensing, pattern recognition and

statistical models.

Alejandro C. Frery graduated in Electronic andElectrical Engineering from the Universidad deMendoza, Argentina. His M.Sc. degree was in Ap-plied Mathematics (Statistics) from the Instituto deMatematica Pura e Aplicada (Rio de Janeiro) andhis Ph.D. degree was in Applied Computing fromthe Instituto Nacional de Pesquisas Espaciais (SaoJose dos Campos, Brazil). He is currently with theInstituto de Computacao, Universidade Federal deAlagoas, Maceio, Brazil. His research interests arestatistical computing and stochastic modeling.

Renato J. Cintra earned his B.Sc., M.Sc., and D.Sc.degrees in Electrical Engineering from UniversidadeFederal de Pernambuco, Brazil, in 1999, 2001, and2005, respectively. In 2005, he joined the Depart-ment of Statistics at UFPE. During 2008-2009, heworked at the University of Calgary, Canada, asa visiting research fellow. He is also a graduatefaculty member of the Department of Electrical andComputer Engineering, University of Akron, OH.His long term topics of research include theory andmethods for digital signal processing, communica-

tions systems, and applied mathematics.